Why is the category of posets defined as it is?

Question

When defining the category of all posets, one says that morphisms are order-preserving functions. However, one can allow order-reversing functions too, since composition is well-behaved (composition of monotone functions is monotone, regardless they preserve or reverse order).

I was wondering if there is any historical/theoretical reason behind.

Answer 1

Note that order-preserving-ness is a particular instance of a much more general phenomenon: namely, that of a homomorphism between two relational structures. By contrast, order-reversing-ness is really specific to the case of a binary relation - what does "order-reversing" mean in the context of a relation with three inputs?

The "specialize-from-relational-structures" theme also lets us cook up the category you have in mind. Given a poset $\mathcal{P}=(P;\le)$, consider the relations $R(x,y,z,w)$ and $C(a,b)$ on $P$ given by $$R=\{(x,y,z,w): (x\le y\leftrightarrow z\le w)\wedge (y\le x\leftrightarrow w\le z)\}$$ and $$C=\{(a,b): a\le b\vee b\le a\}.$$

The "forgetting" process $\mathcal{P}\leadsto\mathcal{P}^-$ gives rise to a category whose objects are the "$\{R,C\}$-structures" coming from posets and whose morphisms are "$\{R,C\}$-preserving" maps.

On a technical level, we might worry that this resulting category is somehow more complicated than the original category of posets it comes from, but in an important sense it isn't: $R$ and $C$ are built from $\le$ in a "nicely-definable" way, and this together with some other more technical observations tells us that at least from the perspective of model theory everything is reasonably nice here. I'm not going to go into detail about this here, but if you're interested look up the notion of "model-theoretic reducts" - the introduction of Junker/Ziegler is a good starting point, and also helps motivate the general study of such processes beyond this particular example.

Crucially, this "poset-derived" category is just the category you have in mind but renamed. That is:

If $\mathcal{P},\mathcal{Q}$ are posets and $f:P\rightarrow Q$ is a function on their underlying sets, then the following are equivalent:

• $f$ is the underlying-set-function of a homomorphism $\mathcal{P}^-\rightarrow\mathcal{Q}^-$.

• $f$ is either order-preserving or order-reversing.

All of this is decent evidence, in my opinion, that the classical idea that homomorphisms should preserve relations is "good enough" - even when we want something broader, such as your category, we can often fit it into this framework in a clear and simple way.